In-Class Exercise 7

In this exercise, we will explore Calibrating Hedonic Pricing Models for Private Highrise Property using the Geographically Weighted Regression (GWR) Method, focusing on spatially varying relationships in property pricing data.

Author

Teng Kok Wai (Walter)

Published

October 14, 2024

Modified

October 16, 2024

1.1 Exercise Reference

Calibrating Hedonic Pricing Model for Private Highrise Property with GWR Method – ISSS626 Geospatial Analytics and Applications

1.2 Overview

In this exercise, we will explore Calibrating Hedonic Pricing Models for Private Highrise Property using the Geographically Weighted Regression (GWR) Method, focusing on spatially varying relationships in property pricing data.

1.3 Learning Outcome

  • Import and load R packages for spatial and statistical analysis.
  • Perform correlation analysis using the ggstatsplot package.
  • Build a hedonic pricing model using Multiple Linear Regression (MLR).
  • Assess and diagnose the MLR model using the olsrr package.
  • Test for spatial autocorrelation in model residuals.
  • Build GWR models using fixed and adaptive bandwidth methods.
  • Visualize GWR outputs, including local R² values and coefficient estimates.
  • Interpret spatial patterns in GWR model results.

1.4 Import the R Packages

The following R packages will be used in this exercise:

Package Purpose Use Case in Exercise
olsrr Provides tools for OLS regression diagnostics and variable selection. Assessing and improving the multiple linear regression model.
ggstatsplot Enhances data visualization and statistical analysis. Performing correlation analysis and visualizing model parameters.
ggpubr Creates publication-ready plots based on ggplot2. Visualizing statistical plots.
sf Handles spatial data operations for vector data. Importing and managing geospatial data like planning subzone boundaries.
spdep Analyzes spatial dependence and weights. Computing spatial weights and performing Moran’s I test.
GWmodel Implements Geographically Weighted Models. Building GWR models with fixed and adaptive bandwidths.
tmap Generates thematic maps. Visualizing spatial data and GWR model outputs.
tidyverse A suite of packages for data manipulation and visualization. Data manipulation and joining datasets.
gtsummary Summarizes data and statistical models. Summarizing regression outputs.
performance Assesses model quality and performance. Evaluating model diagnostics.
see Provides visualization tools for model diagnostics. Visualizing diagnostic plots.
sfdep Analyzes spatial dependence in sf objects. Performing spatial autocorrelation tests on spatial data frames.

To install and load these packages, use the following code:

pacman::p_load(olsrr, ggstatsplot, ggpubr, 
               sf, spdep, GWmodel, tmap,
               tidyverse, gtsummary, performance,
               see, sfdep)

1.5 The Data

The following datasets will be used in this exercise:

Dataset Name Description Format
Master Plan 2014 Subzone Boundary Geospatial data representing the boundaries of different areas in Singapore, specifically at the planning subzone level. ESRI Shapefile
condo_resale_2015 Aspatial data containing records of condominium resale history in Singapore for the year 2015. CSV

2 Importing the data

2.0.1 Importing Geospatial Data

To import MP_SUBZONE_WEB_PL shapefile:

mpsz = st_read(dsn = "data/geospatial", layer = "MP14_SUBZONE_WEB_PL")
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `/Users/walter/code/isss626/isss626-gaa/In-class_Ex/In-class_Ex07/data/geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

The output above shows that the R object used to contain the imported MP14_SUBZONE_WEB_PL shapefile is called mpsz and it is a simple feature object. The geometry type is multipolygon. It is also important to note that mpsz simple feature object does not have EPSG information.

2.0.2 Updating CRS Information

The code below updates the newly imported mpsz with the correct ESPG code (i.e. 3414)

mpsz_svy21 <- st_transform(mpsz, 3414)
st_crs(mpsz_svy21)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

Notice that the EPSG: is indicated as 3414 now.

2.0.3 Importing Aspatial Data

The condo_resale_2015 is in csv file format. The codes chunk below uses read_csv() function of readr package to import condo_resale_2015 into R as a tibble data frame called condo_resale.

condo_resale = read_csv("data/aspatial/Condo_resale_2015.csv")
glimpse(condo_resale)
Rows: 1,436
Columns: 23
$ LATITUDE             <dbl> 1.287145, 1.328698, 1.313727, 1.308563, 1.321437,…
$ LONGITUDE            <dbl> 103.7802, 103.8123, 103.7971, 103.8247, 103.9505,…
$ POSTCODE             <dbl> 118635, 288420, 267833, 258380, 467169, 466472, 3…
$ SELLING_PRICE        <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1320…
$ AREA_SQM             <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 168,…
$ AGE                  <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22, 6,…
$ PROX_CBD             <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783402…
$ PROX_CHILDCARE       <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543, 0…
$ PROX_ELDERLYCARE     <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.121…
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.410632,…
$ PROX_HAWKER_MARKET   <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969, 0…
$ PROX_KINDERGARTEN    <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076, 0…
$ PROX_MRT             <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.528…
$ PROX_PARK            <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.116…
$ PROX_PRIMARY_SCH     <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.709…
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.709…
$ PROX_SHOPPING_MALL   <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.307…
$ PROX_SUPERMARKET     <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.581…
$ PROX_BUS_STOP        <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340, 0…
$ NO_Of_UNITS          <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34, 3…
$ FAMILY_FRIENDLY      <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0…
$ FREEHOLD             <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR       <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…

To display summary statistics of condo_resale:

summary(condo_resale)
    LATITUDE       LONGITUDE        POSTCODE      SELLING_PRICE     
 Min.   :1.240   Min.   :103.7   Min.   : 18965   Min.   :  540000  
 1st Qu.:1.309   1st Qu.:103.8   1st Qu.:259849   1st Qu.: 1100000  
 Median :1.328   Median :103.8   Median :469298   Median : 1383222  
 Mean   :1.334   Mean   :103.8   Mean   :440439   Mean   : 1751211  
 3rd Qu.:1.357   3rd Qu.:103.9   3rd Qu.:589486   3rd Qu.: 1950000  
 Max.   :1.454   Max.   :104.0   Max.   :828833   Max.   :18000000  
    AREA_SQM          AGE           PROX_CBD       PROX_CHILDCARE    
 Min.   : 34.0   Min.   : 0.00   Min.   : 0.3869   Min.   :0.004927  
 1st Qu.:103.0   1st Qu.: 5.00   1st Qu.: 5.5574   1st Qu.:0.174481  
 Median :121.0   Median :11.00   Median : 9.3567   Median :0.258135  
 Mean   :136.5   Mean   :12.14   Mean   : 9.3254   Mean   :0.326313  
 3rd Qu.:156.0   3rd Qu.:18.00   3rd Qu.:12.6661   3rd Qu.:0.368293  
 Max.   :619.0   Max.   :37.00   Max.   :19.1804   Max.   :3.465726  
 PROX_ELDERLYCARE  PROX_URA_GROWTH_AREA PROX_HAWKER_MARKET PROX_KINDERGARTEN 
 Min.   :0.05451   Min.   :0.2145       Min.   :0.05182    Min.   :0.004927  
 1st Qu.:0.61254   1st Qu.:3.1643       1st Qu.:0.55245    1st Qu.:0.276345  
 Median :0.94179   Median :4.6186       Median :0.90842    Median :0.413385  
 Mean   :1.05351   Mean   :4.5981       Mean   :1.27987    Mean   :0.458903  
 3rd Qu.:1.35122   3rd Qu.:5.7550       3rd Qu.:1.68578    3rd Qu.:0.578474  
 Max.   :3.94916   Max.   :9.1554       Max.   :5.37435    Max.   :2.229045  
    PROX_MRT         PROX_PARK       PROX_PRIMARY_SCH  PROX_TOP_PRIMARY_SCH
 Min.   :0.05278   Min.   :0.02906   Min.   :0.07711   Min.   :0.07711     
 1st Qu.:0.34646   1st Qu.:0.26211   1st Qu.:0.44024   1st Qu.:1.34451     
 Median :0.57430   Median :0.39926   Median :0.63505   Median :1.88213     
 Mean   :0.67316   Mean   :0.49802   Mean   :0.75471   Mean   :2.27347     
 3rd Qu.:0.84844   3rd Qu.:0.65592   3rd Qu.:0.95104   3rd Qu.:2.90954     
 Max.   :3.48037   Max.   :2.16105   Max.   :3.92899   Max.   :6.74819     
 PROX_SHOPPING_MALL PROX_SUPERMARKET PROX_BUS_STOP       NO_Of_UNITS    
 Min.   :0.0000     Min.   :0.0000   Min.   :0.001595   Min.   :  18.0  
 1st Qu.:0.5258     1st Qu.:0.3695   1st Qu.:0.098356   1st Qu.: 188.8  
 Median :0.9357     Median :0.5687   Median :0.151710   Median : 360.0  
 Mean   :1.0455     Mean   :0.6141   Mean   :0.193974   Mean   : 409.2  
 3rd Qu.:1.3994     3rd Qu.:0.7862   3rd Qu.:0.220466   3rd Qu.: 590.0  
 Max.   :3.4774     Max.   :2.2441   Max.   :2.476639   Max.   :1703.0  
 FAMILY_FRIENDLY     FREEHOLD      LEASEHOLD_99YR  
 Min.   :0.0000   Min.   :0.0000   Min.   :0.0000  
 1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:0.0000  
 Median :0.0000   Median :0.0000   Median :0.0000  
 Mean   :0.4868   Mean   :0.4227   Mean   :0.4882  
 3rd Qu.:1.0000   3rd Qu.:1.0000   3rd Qu.:1.0000  
 Max.   :1.0000   Max.   :1.0000   Max.   :1.0000  

2.0.4 Converting an Aspatial Data Frame to an sf Object

In this step, we will:

  • Use the st_as_sf() function from the sf package to convert the aspatial data frame into a spatial (sf) object.
  • Apply st_transform() to reproject the coordinates from the WGS 84 coordinate system (CRS: 4326) to SVY21 (CRS: 3414), commonly used in Singapore.
condo_resale_sf <- st_as_sf(condo_resale,
                            coords = c("LONGITUDE", "LATITUDE"),
                            crs = 4326) %>%
  st_transform(crs = 3414)

head(condo_resale_sf)
Simple feature collection with 6 features and 21 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 22085.12 ymin: 29951.54 xmax: 41042.56 ymax: 34546.2
Projected CRS: SVY21 / Singapore TM
# A tibble: 6 × 22
  POSTCODE SELLING_PRICE AREA_SQM   AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
     <dbl>         <dbl>    <dbl> <dbl>    <dbl>          <dbl>            <dbl>
1   118635       3000000      309    30     7.94          0.166            2.52 
2   288420       3880000      290    32     6.61          0.280            1.93 
3   267833       3325000      248    33     6.90          0.429            0.502
4   258380       4250000      127     7     4.04          0.395            1.99 
5   467169       1400000      145    28    11.8           0.119            1.12 
6   466472       1320000      139    22    10.3           0.125            0.789
# ℹ 15 more variables: PROX_URA_GROWTH_AREA <dbl>, PROX_HAWKER_MARKET <dbl>,
#   PROX_KINDERGARTEN <dbl>, PROX_MRT <dbl>, PROX_PARK <dbl>,
#   PROX_PRIMARY_SCH <dbl>, PROX_TOP_PRIMARY_SCH <dbl>,
#   PROX_SHOPPING_MALL <dbl>, PROX_SUPERMARKET <dbl>, PROX_BUS_STOP <dbl>,
#   NO_Of_UNITS <dbl>, FAMILY_FRIENDLY <dbl>, FREEHOLD <dbl>,
#   LEASEHOLD_99YR <dbl>, geometry <POINT [m]>

2.1 Correlation Analysis - ggstatsplot methods

Instead of using corrplot package, in the code chunk below, ggcorrmat() of ggstatsplot is used.

ggcorrmat(condo_resale[, 5:23])

Note

Observations:

  • Some strong positive correlations that is statistically significant includes Proximity to Bus Stops (PROX_BUS_STOP) with Proximity to Childcare (PROX_CHILDCARE) (0.77) and Proximity to Childcare (PROX_CHILDCARE) with Proximity to Primary School (PROX_PRIMARY_SCHOOL) (0.63) among many others.

2.2 Building a Hedonic Pricing Model by using Multiple Linear Regression Method

The code block below using lm() to calibrate the multiple linear regression model.

condo_mlr <- lm(formula = SELLING_PRICE ~ AREA_SQM + 
                  AGE   + PROX_CBD + PROX_CHILDCARE + 
                  PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
                  PROX_HAWKER_MARKET    + PROX_KINDERGARTEN + 
                  PROX_MRT  + PROX_PARK + PROX_PRIMARY_SCH + 
                  PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + 
                  PROX_SUPERMARKET + PROX_BUS_STOP + 
                  NO_Of_UNITS + FAMILY_FRIENDLY + 
                  FREEHOLD + LEASEHOLD_99YR, 
                data=condo_resale_sf)
summary(condo_mlr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + 
    PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + 
    PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
    PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD + 
    LEASEHOLD_99YR, data = condo_resale_sf)

Residuals:
     Min       1Q   Median       3Q      Max 
-3471036  -286903   -22426   239412 12254549 

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)           543071.4   136210.9   3.987 7.03e-05 ***
AREA_SQM               12688.7      370.1  34.283  < 2e-16 ***
AGE                   -24566.0     2766.0  -8.881  < 2e-16 ***
PROX_CBD              -78122.0     6791.4 -11.503  < 2e-16 ***
PROX_CHILDCARE       -333219.0   111020.3  -3.001 0.002734 ** 
PROX_ELDERLYCARE      170950.0    42110.8   4.060 5.19e-05 ***
PROX_URA_GROWTH_AREA   38507.6    12523.7   3.075 0.002147 ** 
PROX_HAWKER_MARKET     23801.2    29299.9   0.812 0.416739    
PROX_KINDERGARTEN     144098.0    82738.7   1.742 0.081795 .  
PROX_MRT             -322775.9    58528.1  -5.515 4.14e-08 ***
PROX_PARK             564487.9    66563.0   8.481  < 2e-16 ***
PROX_PRIMARY_SCH      186170.5    65515.2   2.842 0.004553 ** 
PROX_TOP_PRIMARY_SCH    -477.1    20598.0  -0.023 0.981525    
PROX_SHOPPING_MALL   -207721.5    42855.5  -4.847 1.39e-06 ***
PROX_SUPERMARKET      -48074.7    77145.3  -0.623 0.533273    
PROX_BUS_STOP         675755.0   138552.0   4.877 1.20e-06 ***
NO_Of_UNITS             -216.2       90.3  -2.394 0.016797 *  
FAMILY_FRIENDLY       142128.3    47055.1   3.020 0.002569 ** 
FREEHOLD              300646.5    77296.5   3.890 0.000105 ***
LEASEHOLD_99YR        -77137.4    77570.9  -0.994 0.320192    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 755800 on 1416 degrees of freedom
Multiple R-squared:  0.652, Adjusted R-squared:  0.6474 
F-statistic: 139.6 on 19 and 1416 DF,  p-value: < 2.2e-16
class(condo_mlr)
[1] "lm"

The output is an lm object containing various important pieces of information:

  • Residuals: The differences between the actual and predicted property prices. These will be useful later for spatial analysis to check for any spatial autocorrelation or patterns in the model’s errors.

  • Fitted Values: The predicted selling prices based on the model’s coefficients. These represent the estimated prices of the properties given their characteristics, such as area, proximity to amenities, and tenure type.

2.3 Model Assessment: olsrr method

In this section, we will use olsrr to perform OLS regression. It provides a collection of very useful methods for building better multiple linear regression models:

  • comprehensive regression output
  • residual diagnostics
  • measures of influence
  • heteroskedasticity tests
  • model fit assessment
  • variable contribution assessment
  • variable selection procedures

2.3.1 Generating tidy linear regression report

ols_regress(condo_mlr)
                                Model Summary                                 
-----------------------------------------------------------------------------
R                            0.807       RMSE                     750537.537 
R-Squared                    0.652       MSE                571262902261.220 
Adj. R-Squared               0.647       Coef. Var                    43.160 
Pred R-Squared               0.637       AIC                       42971.173 
MAE                     412117.987       SBC                       43081.835 
-----------------------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 
 AIC: Akaike Information Criteria 
 SBC: Schwarz Bayesian Criteria 

                                     ANOVA                                       
--------------------------------------------------------------------------------
                    Sum of                                                      
                   Squares          DF         Mean Square       F         Sig. 
--------------------------------------------------------------------------------
Regression    1.515738e+15          19        7.977571e+13    139.648    0.0000 
Residual      8.089083e+14        1416    571262902261.220                      
Total         2.324647e+15        1435                                          
--------------------------------------------------------------------------------

                                               Parameter Estimates                                                
-----------------------------------------------------------------------------------------------------------------
               model           Beta    Std. Error    Std. Beta       t        Sig           lower          upper 
-----------------------------------------------------------------------------------------------------------------
         (Intercept)     543071.420    136210.918                   3.987    0.000     275874.535     810268.305 
            AREA_SQM      12688.669       370.119        0.579     34.283    0.000      11962.627      13414.710 
                 AGE     -24566.001      2766.041       -0.166     -8.881    0.000     -29991.980     -19140.022 
            PROX_CBD     -78121.985      6791.377       -0.267    -11.503    0.000     -91444.227     -64799.744 
      PROX_CHILDCARE    -333219.036    111020.303       -0.087     -3.001    0.003    -551000.984    -115437.089 
    PROX_ELDERLYCARE     170949.961     42110.748        0.083      4.060    0.000      88343.803     253556.120 
PROX_URA_GROWTH_AREA      38507.622     12523.661        0.059      3.075    0.002      13940.700      63074.545 
  PROX_HAWKER_MARKET      23801.197     29299.923        0.019      0.812    0.417     -33674.725      81277.120 
   PROX_KINDERGARTEN     144097.972     82738.669        0.030      1.742    0.082     -18205.570     306401.514 
            PROX_MRT    -322775.874     58528.079       -0.123     -5.515    0.000    -437586.937    -207964.811 
           PROX_PARK     564487.876     66563.011        0.148      8.481    0.000     433915.162     695060.590 
    PROX_PRIMARY_SCH     186170.524     65515.193        0.072      2.842    0.005      57653.253     314687.795 
PROX_TOP_PRIMARY_SCH       -477.073     20597.972       -0.001     -0.023    0.982     -40882.894      39928.747 
  PROX_SHOPPING_MALL    -207721.520     42855.500       -0.109     -4.847    0.000    -291788.613    -123654.427 
    PROX_SUPERMARKET     -48074.679     77145.257       -0.012     -0.623    0.533    -199405.956     103256.599 
       PROX_BUS_STOP     675755.044    138551.991        0.133      4.877    0.000     403965.817     947544.272 
         NO_Of_UNITS       -216.180        90.302       -0.046     -2.394    0.017       -393.320        -39.040 
     FAMILY_FRIENDLY     142128.272     47055.082        0.056      3.020    0.003      49823.107     234433.438 
            FREEHOLD     300646.543     77296.529        0.117      3.890    0.000     149018.525     452274.561 
      LEASEHOLD_99YR     -77137.375     77570.869       -0.030     -0.994    0.320    -229303.551      75028.801 
-----------------------------------------------------------------------------------------------------------------
Note

A quick glance on the report indicates that not all variables are statistically significant, meaning that while some factors strongly influence property prices, and others have minimal impact based on this model.

2.3.1.1 Multicollinearity

ols_vif_tol(condo_mlr)
              Variables Tolerance      VIF
1              AREA_SQM 0.8601326 1.162611
2                   AGE 0.7011585 1.426211
3              PROX_CBD 0.4575471 2.185567
4        PROX_CHILDCARE 0.2898233 3.450378
5      PROX_ELDERLYCARE 0.5922238 1.688551
6  PROX_URA_GROWTH_AREA 0.6614081 1.511926
7    PROX_HAWKER_MARKET 0.4373874 2.286303
8     PROX_KINDERGARTEN 0.8356793 1.196631
9              PROX_MRT 0.4949877 2.020252
10            PROX_PARK 0.8015728 1.247547
11     PROX_PRIMARY_SCH 0.3823248 2.615577
12 PROX_TOP_PRIMARY_SCH 0.4878620 2.049760
13   PROX_SHOPPING_MALL 0.4903052 2.039546
14     PROX_SUPERMARKET 0.6142127 1.628100
15        PROX_BUS_STOP 0.3311024 3.020213
16          NO_Of_UNITS 0.6543336 1.528272
17      FAMILY_FRIENDLY 0.7191719 1.390488
18             FREEHOLD 0.2728521 3.664990
19       LEASEHOLD_99YR 0.2645988 3.779307
Note

Even though FREEHOLD and LEASEHOLD_99YR are highly correlated, we don’t need to remove either variable because their Variance Inflation Factor (VIF) values are both less than 5. This indicates that while they are correlated, they do not introduce significant multicollinearity that would negatively impact the stability of the regression model.

Recap on VIF:

The Variance Inflation Factor (VIF) measures the extent of multicollinearity in a regression model. Specifically, it quantifies how much the variance of a regression coefficient is inflated due to collinearity with other predictors. A VIF < 5 generally suggests that multicollinearity is not severe, while values above 10 indicate a high degree of collinearity that could distort the model’s results.

2.3.2 Variable Selection

In this section, we are performing automatic parameter selection based on statistical significance, ensuring that only the most relevant variables are included in the model.

Variable Selection Methods:

  1. Forward Stepwise: Variables are added one by one, starting with the most statistically significant.
  2. Backward Stepwise: Variables are removed one by one, starting with the least significant.
  3. Mixed Stepwise: A combination of forward and backward stepwise, where variables can be added or removed at each step.

We typically use forward stepwise selection with a p-value threshold (forward_p) since we want to retain variables that are statistically significant at each step of the model building process.

condo_fw_mlr <- ols_step_forward_p(
  condo_mlr,
  p_val = 0.05,  # Only include variables with p-value < 0.05
  details = FALSE # Set to TRUE to display the output at each step
)
plot(condo_fw_mlr)

2.3.3 Visualising Model Parameters

In this section, we use the ggcoefstats function from the ggstatsplot package to visualize the coefficients of the regression model. This plot helps in interpreting the magnitude and direction of the relationships between the predictor variables and the dependent variable (selling price).

ggcoefstats(condo_mlr,
            sort = "ascending")

2.3.4 Test for Non-Linearity

In multiple linear regression, it is important for us to test the assumption that linearity and additivity of the relationship between dependent and independent variables.

In the code block below, the ols_plot_resid_fit() of olsrr package is used to perform linearity assumption test.

ols_plot_resid_fit(condo_fw_mlr$model)

Note

Observations:

The figure above reveals that most of the data poitns are scattered around the 0 line, hence we can safely conclude that the relationships between the dependent variable and independent variables are linear.

2.3.5 Test for Normality Assumption

Lastly, the code block below uses ols_plot_resid_hist() of olsrr package to perform normality assumption test.

ols_plot_resid_hist(condo_fw_mlr$model)

Note

Observations:

The figure reveals that the residual of the multiple linear regression model (i.e. condo.mlr1) resembles normal distribution.

When formal statistical test methods is preferred, we can use ols_test_normality() of olsrr package as shown in the code block below.

ols_test_normality(condo_fw_mlr$model)
-----------------------------------------------
       Test             Statistic       pvalue  
-----------------------------------------------
Shapiro-Wilk              0.6856         0.0000 
Kolmogorov-Smirnov        0.1366         0.0000 
Cramer-von Mises         121.0768        0.0000 
Anderson-Darling         67.9551         0.0000 
-----------------------------------------------
Note

Observations:

The summary table above reveals that the p-values of the four tests are way smaller than the alpha value of 0.05. Hence we will reject the null hypothesis and infer that there is statistical evidence that the residual are not normally distributed.

2.4 Testing for Spatial Autocorrelation

The hedonic model we try to build are using geographically referenced attributes, hence it is also important for us to visualize the residual of the hedonic pricing model.

First, we will export the residual of the hedonic pricing model and save it as a data frame. We will also simplify the variable names for easy reference and usage.

mlr_output <- as.data.frame(condo_fw_mlr$model$residuals) %>%
  rename(`FW_MLR_RES` = `condo_fw_mlr$model$residuals`)

Next, we will join the newly created data frame with condo_resale_sf object.

condo_resale_sf <- cbind(condo_resale_sf, 
                        mlr_output$FW_MLR_RES) %>%
  rename(`MLR_RES` = `mlr_output.FW_MLR_RES`)

Next, we will use tmap package to display the distribution of the residuals on an interactive map.

tmap_mode("view")
tm_shape(mpsz)+
  tmap_options(check.and.fix = TRUE) +
  tm_polygons(alpha = 0.4) +
tm_shape(condo_resale_sf) +  
  tm_dots(col = "MLR_RES",
          alpha = 0.6,
          style="quantile")
tmap_mode("plot")
Note

Observations:

The figure above reveal that there is sign of spatial autocorrelation.

2.4.1 Spatial Stationary Test

To proof that our observation is indeed true, we will perform the Moran’s I test.

\(H_o\): The residuals are randomly distributed (also known as spatial stationary) .

\(H_1\): The residuals are spatially non-stationary.

First, we will compute the distance-based weight matrix by using dnearneigh() function of spdep.

condo_resale_sf <- condo_resale_sf %>%
  mutate(nb = st_knn(geometry, k=6,
                     longlat = FALSE),
         wt = st_weights(nb,
                         style = "W"),
         .before = 1)

Next, global_moran_perm() of sfdep is used to perform global Moran permutation test.

# for reproducibility
set.seed(1234) 

global_moran_perm(condo_resale_sf$MLR_RES, 
                  condo_resale_sf$nb, 
                  condo_resale_sf$wt, 
                  alternative = "two.sided", 
                  nsim = 99)

    Monte-Carlo simulation of Moran I

data:  x 
weights: listw  
number of simulations + 1: 100 

statistic = 0.32254, observed rank = 100, p-value < 2.2e-16
alternative hypothesis: two.sided
Note

Observations:

The Global Moran’s I test for residual spatial autocorrelation shows that it’s p-value is less than 2.2e-16 which is less than the alpha value of 0.05. Hence, we will reject the null hypothesis that the residuals are randomly distributed.

Since the Observed Global Moran I = 0.32254 which is greater than 0, we can infer than the residuals resemble cluster distribution.

2.5 Building Hedonic Pricing Models using GWmodel

In this section, we will model hedonic pricing by using geographically weighted regression model. Two spatial weights will be used, they are: fixed and adaptive bandwidth schemes.

2.5.1 Building Fixed Bandwidth GWR Model

2.5.1.1 Computing fixed bandwith

In the code block below bw.gwr() of GWModel package is used to determine the optimal fixed bandwidth to use in the model. Notice that the argument adaptive is set to FALSE indicates that we are interested to compute the fixed bandwidth.

There are two possible approaches can be uused to determine the stopping rule, they are: CV cross-validation approach and AIC corrected (AICc) approach. We define the stopping rule using approach agreement.

bw_fixed <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                     PROX_CBD + PROX_CHILDCARE + 
                     PROX_ELDERLYCARE   + PROX_URA_GROWTH_AREA + 
                     PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH + 
                     PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                     NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                   data=condo_resale_sf, 
                   approach="CV", 
                   kernel="gaussian", 
                   adaptive=FALSE, 
                   longlat=FALSE)
Fixed bandwidth: 17660.96 CV score: 8.259118e+14 
Fixed bandwidth: 10917.26 CV score: 7.970454e+14 
Fixed bandwidth: 6749.419 CV score: 7.273273e+14 
Fixed bandwidth: 4173.553 CV score: 6.300006e+14 
Fixed bandwidth: 2581.58 CV score: 5.404958e+14 
Fixed bandwidth: 1597.687 CV score: 4.857515e+14 
Fixed bandwidth: 989.6077 CV score: 4.722431e+14 
Fixed bandwidth: 613.7939 CV score: 1.379526e+16 
Fixed bandwidth: 1221.873 CV score: 4.778717e+14 
Fixed bandwidth: 846.0596 CV score: 4.791629e+14 
Fixed bandwidth: 1078.325 CV score: 4.751406e+14 
Fixed bandwidth: 934.7772 CV score: 4.72518e+14 
Fixed bandwidth: 1023.495 CV score: 4.730305e+14 
Fixed bandwidth: 968.6643 CV score: 4.721317e+14 
Fixed bandwidth: 955.7206 CV score: 4.722072e+14 
Fixed bandwidth: 976.6639 CV score: 4.721387e+14 
Fixed bandwidth: 963.7202 CV score: 4.721484e+14 
Fixed bandwidth: 971.7199 CV score: 4.721293e+14 
Fixed bandwidth: 973.6083 CV score: 4.721309e+14 
Fixed bandwidth: 970.5527 CV score: 4.721295e+14 
Fixed bandwidth: 972.4412 CV score: 4.721296e+14 
Fixed bandwidth: 971.2741 CV score: 4.721292e+14 
Fixed bandwidth: 970.9985 CV score: 4.721293e+14 
Fixed bandwidth: 971.4443 CV score: 4.721292e+14 
Fixed bandwidth: 971.5496 CV score: 4.721293e+14 
Fixed bandwidth: 971.3793 CV score: 4.721292e+14 
Fixed bandwidth: 971.3391 CV score: 4.721292e+14 
Fixed bandwidth: 971.3143 CV score: 4.721292e+14 
Fixed bandwidth: 971.3545 CV score: 4.721292e+14 
Fixed bandwidth: 971.3296 CV score: 4.721292e+14 
Fixed bandwidth: 971.345 CV score: 4.721292e+14 
Fixed bandwidth: 971.3355 CV score: 4.721292e+14 
Fixed bandwidth: 971.3413 CV score: 4.721292e+14 
Fixed bandwidth: 971.3377 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 
Fixed bandwidth: 971.3396 CV score: 4.721292e+14 
Fixed bandwidth: 971.3402 CV score: 4.721292e+14 
Fixed bandwidth: 971.3398 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
Fixed bandwidth: 971.3399 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
Note

Observations:

The result shows that the recommended bandwidth is 971.3405 metres. Since we are using CRS 3414 which use meter as unit, the unit is in meter.

2.5.1.2 GWModel method - fixed bandwith

Now we can use the code chunk below to calibrate the gwr model using fixed bandwidth and gaussian kernel.

gwr_fixed <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + 
                         AGE    + PROX_CBD + PROX_CHILDCARE + 
                         PROX_ELDERLYCARE   +PROX_URA_GROWTH_AREA + 
                         PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH +
                         PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                         NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                       data=condo_resale_sf, 
                       bw=bw_fixed, 
                       kernel = 'gaussian', 
                       longlat = FALSE)

gwr_fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-10-16 02:01:39.452004 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale_sf, bw = bw_fixed, kernel = "gaussian", 
    longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 971.34 
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -3.5988e+07 -5.1998e+05  7.6780e+05  1.7412e+06
   AREA_SQM              1.0003e+03  5.2758e+03  7.4740e+03  1.2301e+04
   AGE                  -1.3475e+05 -2.0813e+04 -8.6260e+03 -3.7784e+03
   PROX_CBD             -7.7047e+07 -2.3608e+05 -8.3599e+04  3.4646e+04
   PROX_CHILDCARE       -6.0097e+06 -3.3667e+05 -9.7426e+04  2.9007e+05
   PROX_ELDERLYCARE     -3.5001e+06 -1.5970e+05  3.1970e+04  1.9577e+05
   PROX_URA_GROWTH_AREA -3.0170e+06 -8.2013e+04  7.0749e+04  2.2612e+05
   PROX_MRT             -3.5282e+06 -6.5836e+05 -1.8833e+05  3.6922e+04
   PROX_PARK            -1.2062e+06 -2.1732e+05  3.5383e+04  4.1335e+05
   PROX_PRIMARY_SCH     -2.2695e+07 -1.7066e+05  4.8472e+04  5.1555e+05
   PROX_SHOPPING_MALL   -7.2585e+06 -1.6684e+05 -1.0517e+04  1.5923e+05
   PROX_BUS_STOP        -1.4676e+06 -4.5207e+04  3.7601e+05  1.1664e+06
   NO_Of_UNITS          -1.3170e+03 -2.4822e+02 -3.0846e+01  2.5496e+02
   FAMILY_FRIENDLY      -2.2749e+06 -1.1140e+05  7.6214e+03  1.6107e+05
   FREEHOLD             -9.2067e+06  3.8074e+04  1.5169e+05  3.7528e+05
                             Max.
   Intercept            112794435
   AREA_SQM                 21575
   AGE                     434203
   PROX_CBD               2704604
   PROX_CHILDCARE         1654086
   PROX_ELDERLYCARE      38867861
   PROX_URA_GROWTH_AREA  78515805
   PROX_MRT               3124325
   PROX_PARK             18122439
   PROX_PRIMARY_SCH       4637517
   PROX_SHOPPING_MALL     1529953
   PROX_BUS_STOP         11342209
   NO_Of_UNITS              12907
   FAMILY_FRIENDLY        1720745
   FREEHOLD               6073642
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 438.3807 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 997.6193 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 42263.61 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41632.36 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 42515.71 
   Residual sum of squares: 2.534069e+14 
   R-square value:  0.8909912 
   Adjusted R-square value:  0.8430418 

   ***********************************************************************
   Program stops at: 2024-10-16 02:01:40.871896 
Note

Observations: The output is saved in a list of class gwrm. The report shows that the AICc of the gwr is 42263.61 which is significantly smaller than the global multiple linear regression model of 42967.1.

2.5.2 Building Adaptive Bandwidth GWR Model

In this section, we will calibrate the gwr-based hedonic pricing model by using adaptive bandwidth approach.

2.5.2.1 Computing the adaptive bandwidth

Similar to the earlier section, we will first use bw.gwr() to determine the recommended data point to use.

The code block used look very similar to the one used to compute the fixed bandwidth except the adaptive argument has changed to TRUE.

bw_adaptive <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE  + 
                        PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE    + 
                        PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                        PROX_PRIMARY_SCH + PROX_SHOPPING_MALL   + PROX_BUS_STOP + 
                        NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                      data=condo_resale_sf, 
                      approach="CV", 
                      kernel="gaussian", 
                      adaptive=TRUE, 
                      longlat=FALSE)
Adaptive bandwidth: 895 CV score: 7.952401e+14 
Adaptive bandwidth: 561 CV score: 7.667364e+14 
Adaptive bandwidth: 354 CV score: 6.953454e+14 
Adaptive bandwidth: 226 CV score: 6.15223e+14 
Adaptive bandwidth: 147 CV score: 5.674373e+14 
Adaptive bandwidth: 98 CV score: 5.426745e+14 
Adaptive bandwidth: 68 CV score: 5.168117e+14 
Adaptive bandwidth: 49 CV score: 4.859631e+14 
Adaptive bandwidth: 37 CV score: 4.646518e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 
Adaptive bandwidth: 25 CV score: 4.430816e+14 
Adaptive bandwidth: 32 CV score: 4.505602e+14 
Adaptive bandwidth: 27 CV score: 4.462172e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 
Note

Observations:

The result shows that the 30 is the recommended data points to be used.

2.5.2.2 Constructing the adaptive bandwidth gwr model

Now, we can go ahead to calibrate the gwr-based hedonic pricing model by using adaptive bandwidth and gaussian kernel as shown in the code chunk below.

gwr_adaptive <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                            PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + 
                            PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                            PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                            NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                          data=condo_resale_sf, 
                          bw=bw_adaptive, 
                          kernel = 'gaussian', 
                          adaptive=TRUE, 
                          longlat = FALSE)
gwr_adaptive
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-10-16 02:01:50.529361 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale_sf, bw = bw_adaptive, kernel = "gaussian", 
    adaptive = TRUE, longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Adaptive bandwidth: 30 (number of nearest neighbours)
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -1.3487e+08 -2.4669e+05  7.7928e+05  1.6194e+06
   AREA_SQM              3.3188e+03  5.6285e+03  7.7825e+03  1.2738e+04
   AGE                  -9.6746e+04 -2.9288e+04 -1.4043e+04 -5.6119e+03
   PROX_CBD             -2.5330e+06 -1.6256e+05 -7.7242e+04  2.6624e+03
   PROX_CHILDCARE       -1.2790e+06 -2.0175e+05  8.7158e+03  3.7778e+05
   PROX_ELDERLYCARE     -1.6212e+06 -9.2050e+04  6.1029e+04  2.8184e+05
   PROX_URA_GROWTH_AREA -7.2686e+06 -3.0350e+04  4.5869e+04  2.4613e+05
   PROX_MRT             -4.3781e+07 -6.7282e+05 -2.2115e+05 -7.4593e+04
   PROX_PARK            -2.9020e+06 -1.6782e+05  1.1601e+05  4.6572e+05
   PROX_PRIMARY_SCH     -8.6418e+05 -1.6627e+05 -7.7853e+03  4.3222e+05
   PROX_SHOPPING_MALL   -1.8272e+06 -1.3175e+05 -1.4049e+04  1.3799e+05
   PROX_BUS_STOP        -2.0579e+06 -7.1461e+04  4.1104e+05  1.2071e+06
   NO_Of_UNITS          -2.1993e+03 -2.3685e+02 -3.4699e+01  1.1657e+02
   FAMILY_FRIENDLY      -5.9879e+05 -5.0927e+04  2.6173e+04  2.2481e+05
   FREEHOLD             -1.6340e+05  4.0765e+04  1.9023e+05  3.7960e+05
                            Max.
   Intercept            18758355
   AREA_SQM                23064
   AGE                     13303
   PROX_CBD             11346650
   PROX_CHILDCARE        2892127
   PROX_ELDERLYCARE      2465671
   PROX_URA_GROWTH_AREA  7384059
   PROX_MRT              1186242
   PROX_PARK             2588497
   PROX_PRIMARY_SCH      3381462
   PROX_SHOPPING_MALL   38038564
   PROX_BUS_STOP        12081592
   NO_Of_UNITS              1010
   FAMILY_FRIENDLY       2072414
   FREEHOLD              1813995
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 350.3088 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 1085.691 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 41982.22 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41546.74 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 41914.08 
   Residual sum of squares: 2.528227e+14 
   R-square value:  0.8912425 
   Adjusted R-square value:  0.8561185 

   ***********************************************************************
   Program stops at: 2024-10-16 02:01:52.321114 
Note

Observations:

The report shows that the AICc the adaptive distance gwr is 41982.22 which is even smaller than the AICc of the fixed distance gwr of 42263.61.

2.5.3 Visualising GWR Output

In addition to regression residuals, the output feature class table includes fields for observed and predicted y values, condition number (cond), Local R2, residuals, and explanatory variable coefficients and standard errors:

  • Condition Number: this diagnostic evaluates local collinearity. In the presence of strong local collinearity, results become unstable. Results associated with condition numbers larger than 30, may be unreliable.

  • Local R2: these values range between 0.0 and 1.0 and indicate how well the local regression model fits observed y values. Very low values indicate the local model is performing poorly. Mapping the Local R2 values to see where GWR predicts well and where it predicts poorly may provide clues about important variables that may be missing from the regression model.

  • Predicted: these are the estimated (or fitted) y values 3. computed by GWR.

  • Residuals: to obtain the residual values, the fitted y values are subtracted from the observed y values. Standardized residuals have a mean of zero and a standard deviation of 1. A cold-to-hot rendered map of standardized residuals can be produce by using these values.

  • Coefficient Standard Error: these values measure the reliability of each coefficient estimate. Confidence in those estimates are higher when standard errors are small in relation to the actual coefficient values. Large standard errors may indicate problems with local collinearity.

They are all stored in a SpatialPointsDataFrame or SpatialPolygonsDataFrame object integrated with fit.points, GWR coefficient estimates, y value, predicted values, coefficient standard errors and t-values in its “data” slot in an object called SDF of the output list.

2.5.4 Converting SDF into sf data.frame

To visualise the fields in SDF, we need to first covert it into sf data.frame by using the code chunk below.

gwr_adaptive_output <- as.data.frame(
  gwr_adaptive$SDF) %>%
  select(-c(2:15))
gwr_sf_adaptive <- cbind(condo_resale_sf,
                         gwr_adaptive_output)

Next, glimpse() is used to display the content of condo_resale_sf.adaptive sf data frame.

glimpse(gwr_sf_adaptive)
Rows: 1,436
Columns: 63
$ nb                      <nb> <66, 77, 123, 238, 239, 343>, <21, 162, 163, 19…
$ wt                      <list> <0.1666667, 0.1666667, 0.1666667, 0.1666667, …
$ POSTCODE                <dbl> 118635, 288420, 267833, 258380, 467169, 466472…
$ SELLING_PRICE           <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ AREA_SQM                <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 1…
$ AGE                     <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22,…
$ PROX_CBD                <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783…
$ PROX_CHILDCARE          <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543…
$ PROX_ELDERLYCARE        <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.…
$ PROX_URA_GROWTH_AREA    <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.4106…
$ PROX_HAWKER_MARKET      <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969…
$ PROX_KINDERGARTEN       <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076…
$ PROX_MRT                <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.…
$ PROX_PARK               <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.…
$ PROX_PRIMARY_SCH        <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.…
$ PROX_TOP_PRIMARY_SCH    <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.…
$ PROX_SHOPPING_MALL      <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.…
$ PROX_SUPERMARKET        <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.…
$ PROX_BUS_STOP           <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340…
$ NO_Of_UNITS             <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34…
$ FAMILY_FRIENDLY         <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0…
$ FREEHOLD                <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR          <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ MLR_RES                 <dbl> -1489099.55, 415494.57, 194129.69, 1088992.71,…
$ Intercept               <dbl> 2050011.67, 1633128.24, 3433608.17, 234358.91,…
$ y                       <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ yhat                    <dbl> 2886531.8, 3466801.5, 3616527.2, 5435481.6, 13…
$ residual                <dbl> 113468.16, 413198.52, -291527.20, -1185481.63,…
$ CV_Score                <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ Stud_residual           <dbl> 0.38207013, 1.01433140, -0.83780678, -2.846146…
$ Intercept_SE            <dbl> 516105.5, 488083.5, 963711.4, 444185.5, 211962…
$ AREA_SQM_SE             <dbl> 823.2860, 825.2380, 988.2240, 617.4007, 1376.2…
$ AGE_SE                  <dbl> 5889.782, 6226.916, 6510.236, 6010.511, 8180.3…
$ PROX_CBD_SE             <dbl> 37411.22, 23615.06, 56103.77, 469337.41, 41064…
$ PROX_CHILDCARE_SE       <dbl> 319111.1, 299705.3, 349128.5, 304965.2, 698720…
$ PROX_ELDERLYCARE_SE     <dbl> 120633.34, 84546.69, 129687.07, 127150.69, 327…
$ PROX_URA_GROWTH_AREA_SE <dbl> 56207.39, 76956.50, 95774.60, 470762.12, 47433…
$ PROX_MRT_SE             <dbl> 185181.3, 281133.9, 275483.7, 279877.1, 363830…
$ PROX_PARK_SE            <dbl> 205499.6, 229358.7, 314124.3, 227249.4, 364580…
$ PROX_PRIMARY_SCH_SE     <dbl> 152400.7, 165150.7, 196662.6, 240878.9, 249087…
$ PROX_SHOPPING_MALL_SE   <dbl> 109268.8, 98906.8, 119913.3, 177104.1, 301032.…
$ PROX_BUS_STOP_SE        <dbl> 600668.6, 410222.1, 464156.7, 562810.8, 740922…
$ NO_Of_UNITS_SE          <dbl> 218.1258, 208.9410, 210.9828, 361.7767, 299.50…
$ FAMILY_FRIENDLY_SE      <dbl> 131474.73, 114989.07, 146607.22, 108726.62, 16…
$ FREEHOLD_SE             <dbl> 115954.0, 130110.0, 141031.5, 138239.1, 210641…
$ Intercept_TV            <dbl> 3.9720784, 3.3460017, 3.5629010, 0.5276150, 1.…
$ AREA_SQM_TV             <dbl> 11.614302, 20.087361, 13.247868, 33.577223, 4.…
$ AGE_TV                  <dbl> -1.6154474, -9.3441881, -4.1023685, -15.524301…
$ PROX_CBD_TV             <dbl> -3.22582173, -6.32792021, -4.62353528, 5.17080…
$ PROX_CHILDCARE_TV       <dbl> 1.000488185, 1.471786337, -0.344047555, 1.5766…
$ PROX_ELDERLYCARE_TV     <dbl> -3.26126929, 3.84626245, 4.13191383, 2.4756745…
$ PROX_URA_GROWTH_AREA_TV <dbl> -2.846248368, -1.848971738, -2.648105057, -5.6…
$ PROX_MRT_TV             <dbl> -1.61864578, -8.92998600, -3.40075727, -7.2870…
$ PROX_PARK_TV            <dbl> -0.83749312, 2.28192684, 0.66565951, -3.340617…
$ PROX_PRIMARY_SCH_TV     <dbl> 1.59230221, 6.70194543, 2.90580089, 12.9836104…
$ PROX_SHOPPING_MALL_TV   <dbl> 2.753588422, -0.886626400, -1.056869486, -0.16…
$ PROX_BUS_STOP_TV        <dbl> 2.0154464, 4.4941192, 3.0419145, 12.8383775, 0…
$ NO_Of_UNITS_TV          <dbl> 0.480589953, -1.380026395, -0.045279967, -0.44…
$ FAMILY_FRIENDLY_TV      <dbl> -0.06902748, 2.69655779, 0.04058290, 14.312764…
$ FREEHOLD_TV             <dbl> 2.6213469, 3.0452799, 1.1970499, 8.7711485, 1.…
$ Local_R2                <dbl> 0.8846744, 0.8899773, 0.8947007, 0.9073605, 0.…
$ geometry                <POINT [m]> POINT (22085.12 29951.54), POINT (25656.…
$ geometry.1              <POINT [m]> POINT (22085.12 29951.54), POINT (25656.…
summary(gwr_adaptive$SDF$yhat)
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
  171347  1102001  1385528  1751842  1982307 13887901 

2.5.5 Visualising local R2

The code block below is used to create an interactive point symbol map.

Warning

Note that there is an unsolved issue with the mpsz data from the official sources.

To prevent potential errors during mapping, we can use the tmap_options(check.and.fix = TRUE) to automatically corrects any issues with the spatial data during visualization.

tmap_mode("view")
tmap_options(check.and.fix = TRUE)
tm_shape(mpsz)+
  tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf_adaptive) +  
  tm_dots(col = "Local_R2",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))
tmap_mode("plot")

2.5.6 Visualising coefficient estimates

The code chunks below is used to create an interactive point symbol map.

tmap_options(check.and.fix = TRUE)
tmap_mode("view")
AREA_SQM_SE <- tm_shape(mpsz)+
  tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf_adaptive) +  
  tm_dots(col = "AREA_SQM_SE",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

AREA_SQM_TV <- tm_shape(mpsz)+
  tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf_adaptive) +  
  tm_dots(col = "AREA_SQM_TV",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

tmap_arrange(AREA_SQM_SE, AREA_SQM_TV, 
             asp=1, ncol=2,
             sync = TRUE)
tmap_mode("plot")

2.5.6.1 By URA Planning Region

tm_shape(mpsz[mpsz$REGION_N=="CENTRAL REGION", ])+
  tm_polygons()+
tm_shape(gwr_sf_adaptive) + 
  tm_bubbles(col = "Local_R2",
           size = 0.15,
           border.col = "gray60",
           border.lwd = 1)